Lemma 4

Tag 00XW, permanent. Cite it as [Bourbaki, Tag 00XW].

Lie Groups and Lie Algebras, chapter VIII, § 1. The Lie algebra sl(2, k) and its representations

Let E be a non-zero finite dimensional $\mathfrak{s}\mathfrak{l}(2, k)$ -module. Then E has primitive elements.

Since $X_+$ is a nilpotent element of $\mathfrak{s}\mathfrak{l}(2, k),X_{+E}$ is nilpotent. Assume that $X_{+E}^{m-1}\not= 0$ and $X_{+E}^m$ = 0. By Lemma 2,

m(H_E-m+ 1)X_{+E}^{m-1}= [X_{-E}, X_{+E}^m] = 0

and hence the elements of $X_+^{m-1}(E)$ -- $\{0\}$ are primitive.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Corollary, § 1
Theorem 1, § 1

Cites

Lemma 2